Implications

We have the following indirect implication of form equivalence classes:

152 \(\Rightarrow\) 325
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
152 \(\Rightarrow\) 4 Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math.
note-27
note-27
note-27
4 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 325 note-46

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
152:

\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.  (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)

4:

Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

325:

Ramsey's Theorem II: \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see Form 17.)

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